Finite Group Databasebeta

← #12156462

Subgroups — #12156462

Order 1280.

946 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×Q8×C22⋊C4C211yesmaximal, characteristic
2C22×#3967C211yesmaximal, characteristic
3C2×#65311C211yesmaximal, characteristic, complemented
4#10605226C211yesmaximal, characteristic
5#10602925C211yesmaximal, characteristic
6#10602422C211yesmaximal, characteristic, complemented
7#10602421C211yesmaximal, characteristic, complemented
8#10602420C211yesmaximal, characteristic, complemented
9#10600479C211yesmaximal, characteristic, complemented
10#10600476C211yesmaximal, characteristic, complemented
11#10600475C211yesmaximal, characteristic, complemented
12#10600462C211yesmaximal, characteristic, complemented
13#10600304C211yesmaximal, characteristic
14#10600297C211yesmaximal, characteristic
15#10600242C211yesmaximal, characteristic
16C23×C5×Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C23×#168C2211yescharacteristic
21C2×#4016C2211yescharacteristic
22C5×#391—21no—
23C5×#391—21no—
24C5×#391C2211yescharacteristic
25C5×#391C2211yescharacteristic
26C5×#380—21no—
27C5×#380C2211yescharacteristic
28C4×C5×C22⋊C4—21no—
29C4×C5×C22⋊C4C2211yescharacteristic
30C2×#3967—24no—
31C2×#3967C2211yescharacteristic
32C2×#3967C2211yescharacteristic
33C2×#3967C2211yescharacteristic
34C2×#3967C2211yescharacteristic
35C2×#3967C2211yescharacteristic
36C2×#3967C2211yescharacteristic
37C2×#3967C2211yescharacteristic
38C2×#3967C2211yescharacteristic
39#65501C2211yescharacteristic
40C2×C22⋊#168C2211yescharacteristic
41#65311—24no—
42C2×#3940C2211yescharacteristic
43C2×#3940C2211yescharacteristic
44C2×#3939C2211yescharacteristic
45C2×#3939C2211yescharacteristic
46#65235—21no—
47#65231—21no—
48#65225C2211yescharacteristic
49#65217C2211yescharacteristic
50#65217C2211yescharacteristic
51C2×C4⋊#168C2211yescharacteristic
52C2×C4⋊#168C2211yescharacteristic
53#64949—21no—
54#64946—21no—
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60#64744C2211yescharacteristic
61#25957—51nomaximal, Sylow
62C22×C5×Q8—21no—
63C22×C5×Q8—22no—
64C22×C5×Q8—22no—
65C22×C5×Q8—22no—
66C22×C5×Q8—41no—
67C22×C5×Q8—41no—
68C22×C5×Q8Q814yes—
69C22×C5×Q8C2311yescharacteristic
70C22×C5×Q8C2311yescharacteristic
71C23×C4×C5—21no—
72C23×C4×C5C2311yescharacteristic
73C23×Dic5—21no—
74C4×C5×Q8—22no—
75C4×C5×Q8—22no—
76C4×C5×Q8—41no—
77C4×C5×Q8—41no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C22⋊C4—21no—
83C2×C5×C22⋊C4D411yescharacteristic
84C2×C5×C22⋊C4D411yescharacteristic
85C2×C42×C5—21no—
86C2×C42×C5—21no—
87C2×C42×C5C2311yescharacteristic
88C2×C42×C5C2311yescharacteristic
89C2×C22⋊Dic5—21no—
90C2×C4⋊Dic5—22no—
91C2×C4⋊Dic5—21no—
92C2×C4⋊Dic5—41no—
93C2×C4⋊Dic5C2311yescharacteristic
94C2×C4⋊Dic5C2311yescharacteristic
95C2×C4⋊Dic5C2311yescharacteristic
96C2×C4⋊Dic5C2311yescharacteristic
97C2×#683—21no—
98C2×#683—21no—
99C2×#683—21no—
100C2×#683—21no—
101C22×#168—22no—
102C22×#168—22no—
103C22×#168—41no—
104C22×#168C2311yescharacteristic
105C22×#168C2311yescharacteristic
106C22×#168C2311yescharacteristic
107C22×#168C2311yescharacteristic
108#4016—22no—
109#4016—41no—
110C5×#95—21no—
111C5×#95—21no—
112C5×#95—21no—
113C5×#95—21no—
114C5×#95—21no—
115C5×#95D411yescharacteristic
116C5×#95D411yescharacteristic
117#3967—22no—
118#3967—22no—
119#3967—22no—
120#3967—22no—
121#3967—22no—
122#3967—41no—
123#3967—41no—
124#3967D414yes—
125#3963—21no—
126#3963—21no—
127#3963—21no—
128#3963—21no—
129#3963—21no—
130C22⋊#168—22no—
131#3940—22no—
132#3940—41no—
133#3939—22no—
134#3939—41no—
135C4⋊#168—22no—
136C4⋊#168—41no—
137C2×#479—51no—
138C22×#103—51no—
139#2442—51no—
140Q8×C22⋊C4D511yescharacteristic
141#2097—51no—
142#2068—51no—
143#2066—51no—
144#2026—51no—
145#1865—51no—
146#1863—51no—
147#1852—51no—
148#1850—51no—
149#1807—51no—
150#1805—51no—
151#1787—51no—
152C24×C5C2×D411yescharacteristic
153C2×C5×Q8—24no—
154C2×C5×Q8—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8—28no—
159C2×C5×Q8—21no—
160C2×C5×Q8—21no—
161C2×C5×Q8—21no—
162C2×C5×Q8—21no—
163C2×C5×Q8—42no—
164C2×C5×Q8—42no—
165C2×C5×Q8—44no—
166C2×C5×Q8—44no—
167C2×C5×Q8—44no—
168C2×C5×Q8C2×D412yes—
169C2×C5×Q8C2×Q812yes—
170C2×C5×Q8C2×Q812yes—
171C2×C5×Q8#4112yes—
172C22×C4×C5—21no—
173C22×C4×C5—21no—
174C22×C4×C5—21no—
175C22×C4×C5—21no—
176C22×C4×C5—21no—
177C22×C4×C5—24no—
178C22×C4×C5—21no—
179C22×C4×C5—21no—
180C22×C4×C5—21no—
181C22×C4×C5—42no—
182C22×C4×C5—42no—
183C22×C4×C5C2×D411yescharacteristic
184C22×C4×C5C2×D411yescharacteristic
185C22×C4×C5C2×D411yescharacteristic
186C22×C4×C5C2×D411yescharacteristic
187C22×C4×C5C2×D411yescharacteristic
188C22×C4×C5C2×Q812yes—
189C22×C4×C5#4112yes—
190C22×C4×C5C2411yescharacteristic
191C22×Dic5—21no—
192C22×Dic5—21no—
193C22×Dic5—21no—
194C22×Dic5—21no—
195C22×Dic5—42no—
196C22×Dic5—42no—
197C22×Dic5—81no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—42no—
203C5×C4⋊C4—42no—
204C5×C22⋊C4—22no—
205C5×C22⋊C4—22no—
206C5×C22⋊C4—42no—
207C42×C5—21no—
208C42×C5—21no—
209C42×C5—21no—
210C42×C5—21no—
211C42×C5—42no—
212C42×C5—42no—
213C22⋊Dic5—42no—
214C4⋊Dic5—21no—
215C4⋊Dic5—21no—
216C4⋊Dic5—21no—
217C4⋊Dic5—21no—
218C4⋊Dic5—21no—
219C4⋊Dic5—21no—
220C4⋊Dic5—21no—
221C4⋊Dic5—42no—
222C4⋊Dic5C2×D412yes—
223#683—42no—
224#683—42no—
225#683—81no—
226#683—81no—
227C2×#168—21no—
228C2×#168—21no—
229C2×#168—21no—
230C2×#168—21no—
231C2×#168—21no—
232C2×#168—21no—
233C2×#168—21no—
234C2×#168—21no—
235C2×#168—21no—
236C2×#168—21no—
237C2×#168—21no—
238C2×#168—44no—
239C2×#168C2×D412yes—
240C23×Q8C2×D511yescharacteristic
241C23×C8—51no—
242C2×#125—51no—
243C2×C4×Q8C2×D511yescharacteristic
244C2×C4×Q8C2×D511yescharacteristic
245C22×C4⋊C4—51no—
246#479—104no—
247C2×#107—51no—
248C2×#107—51no—
249C2×#106—51no—
250C2×#106—51no—
251C2×C4⋊C8—51no—
252C2×C4⋊C8—51no—
253C2×#103—51no—
254C2×#103—51no—
255C2×#103—51no—
256C2×#103—51no—
257C2×#103—51no—
258C2×#103—51no—
259C2×#103—51no—
260C2×#103—51no—
261C2×#103—104no—
262C2×C22⋊C8—51no—
263#400—101no—
264#398—101no—
265#391—21no—
266#391—21no—
267#391C2×D511yescharacteristic
268#391C2×D511yescharacteristic
269#389—51no—
270#389—51no—
271#385—101no—
272#382—51no—
273#380—21no—
274#380—51no—
275#380—101no—
276#380C2×D511yescharacteristic
277C4×C22⋊C4—21no—
278C4×C22⋊C4C2×D511yescharacteristic
279#340—51no—
280#340—51no—
281#340—51no—
282#340—51no—
283#336—51no—
284#336—51no—
285C23×C5—21no—
286C23×C5—21no—
287C23×C5—21no—
288C23×C5—21no—
289C23×C5C22⋊Q812yes—
290C23×C5#13512yes—
291C23×C5#13712yes—
292C23×C5C22×D411yescharacteristic
293C5×Q8—22no—
294C5×Q8—22no—
295C5×Q8—22no—
296C5×Q8—22no—
297C5×Q8—22no—
298C5×Q8—48no—
299C5×Q8—41no—
300C5×Q8—41no—
301C5×Q8C22⋊Q814yes—
302C2×C4×C5—22no—
303C2×C4×C5—21no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—21no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—22no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5—21no—
315C2×C4×C5—21no—
316C2×C4×C5—21no—
317C2×C4×C5—21no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—21no—
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—21no—
324C2×C4×C5—48no—
325C2×C4×C5—41no—
326C2×C4×C5—41no—
327C2×C4×C5—41no—
328C2×C4×C5—41no—
329C2×C4×C5—41no—
330C2×C4×C5—41no—
331C2×C4×C5—41no—
332C2×C4×C5—41no—
333C2×C4×C5C22⋊Q812yes—
334C2×C4×C5C22×D411yescharacteristic
335C2×C4×C5C22×Q811yescharacteristic
336C2×C4×C5C2×#4111yescharacteristic
337C2×C4×C5C2×#4111yescharacteristic
338C2×C4×C5H(5,𝔽2)11yescharacteristic
339C2×C4×C5H(5,𝔽2)11yescharacteristic
340C2×Dic5—22no—
341C2×Dic5—41no—
342C2×Dic5—41no—
343C2×Dic5—41no—
344C2×Dic5—41no—
345C2×Dic5—41no—
346C2×Dic5—41no—
347C2×Dic5—41no—
348C2×Dic5—41no—
349C2×Dic5—41no—
350C2×Dic5—41no—
351C2×Dic5—41no—
352C2×Dic5—84no—
353#168—22no—
354#168—41no—
355#168—41no—
356#168—41no—
357C22×Q8—22no—
358C22×Q8—21no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—41no—
362C22×Q8—41no—
363C22×Q8Dic1014yes—
364C22×Q8C22×D511yescharacteristic
365C22×Q8C22×D511yescharacteristic
366C23×C4—21no—
367C23×C4—101no—
368C23×C4C22×D511yescharacteristic
369C22×C8—51no—
370C22×C8—51no—
371C22×C8—51no—
372C22×C8—51no—
373C22×C8—102no—
374C22×C8—102no—
375C22×C8—201no—
376#125—102no—
377#125—201no—
378C4×Q8—22no—
379C4×Q8—22no—
380C4×Q8—41no—
381C4×Q8—41no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—51no—
385C2×C4⋊C4—51no—
386C2×C4⋊C4—51no—
387C2×C4⋊C4—51no—
388C2×C4⋊C4—102no—
389C2×C4⋊C4—101no—
390C2×C4⋊C4—101no—
391C2×C4⋊C4—101no—
392C2×C4⋊C4—101no—
393C2×C4⋊C4—101no—
394C2×C4⋊C4—201no—
395C2×C4⋊C4C22×D511yescharacteristic
396C2×C4⋊C4C22×D511yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—101no—
399C2×C22⋊C4C22⋊D511yescharacteristic
400C2×C22⋊C4C22⋊D511yescharacteristic
401C2×C42—21no—
402C2×C42—21no—
403C2×C42C22×D511yescharacteristic
404C2×C42C22×D511yescharacteristic
405#107—102no—
406#107—201no—
407#106—102no—
408#106—201no—
409C4⋊C8—102no—
410C4⋊C8—201no—
411#103—54no—
412#103—102no—
413#103—102no—
414#103—102no—
415#103—102no—
416#103—102no—
417#103—201no—
418#103—201no—
419C22⋊C8—102no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—101no—
426#95—101no—
427#95—101no—
428#95—101no—
429#95—101no—
430#95C22⋊D511yescharacteristic
431#95C22⋊D511yescharacteristic
432C22×C5—22no—
433C22×C5—22no—
434C22×C5—22no—
435C22×C5—22no—
436C22×C5—22no—
437C22×C5—22no—
438C22×C5#47914yescomplemented
439C22×C5C2×C22⋊Q811yescharacteristic
440C22×C5D4211yescharacteristic
441C22×C5#54611yescharacteristic
442C22×C5C2×#13511yescharacteristic
443C22×C5C2×#13711yescharacteristic
444C22×C5#57711yescharacteristic
445C22×C5#57811yescharacteristic
446C4×C5—22no—
447C4×C5—24no—
448C4×C5—42no—
449C4×C5—41no—
450C4×C5—41no—
451C4×C5—41no—
452C4×C5—41no—
453C4×C5—81no—
454C4×C5—81no—
455C4×C5C2×C22⋊Q811yescharacteristic
456C4×C5#54311yescharacteristic
457C4×C5#54611yescharacteristic
458C4×C5#55111yescharacteristic
459Dic5—42no—
460Dic5—81no—
461Dic5—81no—
462Dic5—81no—
463C24C2×C22⋊D511yescharacteristic
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—24no—
469C2×Q8—28no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—44no—
477C2×Q8—44no—
478C2×Q8—44no—
479C2×Q8C2×Dic1012yes—
480C2×Q8C2×Dic1012yes—
481C2×Q8D4×D512yes—
482C2×Q8#71112yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—101no—
495C22×C4—101no—
496C22×C4—101no—
497C22×C4—101no—
498C22×C4—202no—
499C22×C4—202no—
500C22×C4—401no—
501C22×C4C2×Dic1012yes—
502C22×C4#70912yes—
503C22×C4C2×C22⋊D511yescharacteristic
504C22×C4C2×C22⋊D511yescharacteristic
505C22×C4C2×C22⋊D511yescharacteristic
506C22×C4C2×C22⋊D511yescharacteristic
507C22×C4C2×C22⋊D511yescharacteristic
508C22×C4C23×D511yescharacteristic
509C2×C8—52no—
510C2×C8—101no—
511C2×C8—101no—
512C2×C8—101no—
513C2×C8—101no—
514C2×C8—101no—
515C2×C8—101no—
516C2×C8—101no—
517C2×C8—101no—
518C2×C8—101no—
519C2×C8—101no—
520C2×C8—101no—
521C2×C8—204no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—52no—
529C4⋊C4—101no—
530C4⋊C4—101no—
531C4⋊C4—101no—
532C4⋊C4—101no—
533C4⋊C4—101no—
534C4⋊C4—101no—
535C4⋊C4—101no—
536C4⋊C4—202no—
537C4⋊C4—202no—
538C4⋊C4—202no—
539C4⋊C4—401no—
540C4⋊C4—401no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—42no—
544C22⋊C4—202no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C42—42no—
551C2×C5—24no—
552C2×C5#243711yescharacteristic
553C2×C5C2×#47911yescharacteristic
554C2×C5#308211yescharacteristic
555C2×C5#327011yescharacteristic
556C2×C5#328211yescharacteristic
557C2×C5#330011yescharacteristic
558C2×C5#332011yescharacteristic
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23#407012yes—
564C23#408812yes—
565C23#409612yes—
566C23C22×C22⋊D511yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—48no—
573Q8—41no—
574Q8—41no—
575Q8#402414yes—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—102no—
608C2×C4—201no—
609C2×C4—201no—
610C2×C4—201no—
611C2×C4—201no—
612C2×C4—201no—
613C2×C4—201no—
614C2×C4—201no—
615C2×C4—201no—
616C2×C4—201no—
617C2×C4—201no—
618C2×C4—201no—
619C2×C4—404no—
620C2×C4#407012yes—
621C2×C4C22×Dic1011yescharacteristic
622C2×C4C2×#70911yescharacteristic
623C2×C4C2×D4×D511yescharacteristic
624C2×C4C2×#71111yescharacteristic
625C2×C4#414411yescharacteristic
626C2×C4#414411yescharacteristic
627C8—102no—
628C8—201no—
629C8—201no—
630C8—201no—
631C5#2595711yescharacteristic, complemented, Sylow
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#6531114yescomplemented
639C22C2×#407011yescharacteristic
640C22D4×C22⋊D511yescharacteristic
641C22#6614011yescharacteristic
642C22C2×#408811yescharacteristic
643C22C2×#409611yescharacteristic
644C22#6617111yescharacteristic
645C22#6617211yescharacteristic
646C4—22no—
647C4—24no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—202no—
656C4—401no—
657C4—401no—
658C4—401no—
659C4C2×#402411yescharacteristic
660C4#6581711yescharacteristic
661C4#6582411yescharacteristic
662C4#6582911yescharacteristic
663C2—24no—
664C2#1061384111yescharacteristic
665C2C2×#6531111yescharacteristic
666C2#1061428111yescharacteristic
667C2#1061526511yescharacteristic
668C2#1061527611yescharacteristic
669C2#1061528411yescharacteristic
670C2#1061529611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×Q8×C22⋊C4C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenterQ8×C22⋊C4D5
hypocenterC5#25957
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#65311C2semidirect
6. #10602422C2semidirect
7. #10602421C2semidirect
8. #10602420C2semidirect
9. #10600479C2semidirect
10. #10600476C2semidirect
11. #10600475C2semidirect
12. #10600462C2semidirect
438. C22×C5#479semidirect
631. C5#25957semidirect
638. C22#65311semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×Q8×C22⋊C42
C22×#39672
C2×#653112
#106052262
#106029252
#106024222
#106024212
#106024202
#106004792
#106004762
#106004752
#106004622
#106003042
#106002972
#106002422
#259575